Combinatorics

New work establishes a tighter connection between the rank of a polynomial and the extent to which it favors particular outputs.

The n-queens problem is about finding how many different ways queens can be placed on a chessboard so that none attack each other. A mathematician has now all but solved it.

A pair of mathematicians solved a legendary question about the proportion of vertices in a graph with an odd number of connections.

Researchers have proved a special case of the Erdős-Hajnal conjecture, which shows what happens in graphs that exclude anything resembling a pentagon.

Fifty years ago, Paul Erdős and two other mathematicians came up with a graph theory problem that they thought they might solve on the spot. A team of mathematicians has finally settled it.

The mathematician Federico Ardila takes a creative approach to the search for useful answers hiding among inconceivably huge numbers of possible ones.

Avi Wigderson and László Lovász won for their work developing complexity theory and graph theory, respectively, and for connecting the two fields.

Po-Shen Loh has harnessed his competitive impulses and iconoclastic tendencies to reinvigorate the U.S. Math Olympiad program.

Lauren Williams has charted an adventurous mathematical career out of the pieces of a fundamental object called the positive Grassmannian.